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What is the force of the participle ἐνεργουμένη in this verse? Should I read it as passive or middle?

How would the meaning of the sentence be changed if it were simply omitted? What is the difference between these two sentences:

πολὺ ἰσχύει δέησις δικαίου ἐνεργουμένη.

πολὺ ἰσχύει δέησις δικαίου.

iIn traditional terms, I'd call this a circumstantialparticiple, "when it is in action" or "when it is actively at work." The form is pretty clearly middle -- so far as I can judge, there are no passive instances of this verb in the GNT. Note what BDAG says of the middle:

I haven't see the article by Clark arguing for a passive sense referred to by Danker as KClark, The Mng. of ἐνεργέω and καταργέω in the NT: JBL 54, ’35, 93–101.

K. W. Clark's article was reprinted in his The Gentile Bias and Other Essays (1980) 183-91, of which unfortunately Google apparently displays only the first few pages, not the part where he deals with the passive uses. His argument is that the agent of the verb is in the NT always some supernatural spiritual force (God or other), that this fits well in a 1st-century religious context, and it is not taken account of if you read the M-P forms as middle, as many do. Clark also refers to J Armitage Robinson, who in an Excursus in his Ephesians commentary takes the same line, at least for the Pauline instances.

"The passive interpretation being thus supported by the early Greek and Latin commentators, as well as by the constant usage in non-biblical Greek, we are naturally led to ask whether there is any necessity for a different explanation in the nine passages of the N.T. in which the word occurs."
Also: "It is to be observed that in actual meaning ἐνεργεῖν and ἐνεργεῖσθαι come nearly to the same thing. Only the passive serves to remind us that the operation is not self-originated. The powers 'work' indeed; but they 'are made to work'."http://archive.org/stream/MN41514ucmf_2 ... 3/mode/2up

The references given in the Bauer lexicon are not necessarily to be read with the glosses given there. E.g., Diod Sic 13.85.2
οὐ προσδεξαμένων δέ τῶν ἐν τῇ πόλει τούς λόγους, εὐθὺς τὰ τῆς πολιορκίας ἐνηργεῖτο is translated by Oldfather (Loeb ed.) "when the inhabitants of the city would not entertain these terms, the siege was begun at once". (More strictly, I suggest: was put into effect at once.)

Tony, I very much appreciate your full response, references, and especially the reference to the discussion on Betty Bibles Blog regarding interpretation of the middle-passive of ἐνεργεῖς/ἐνεργεῖσθαι in a passive sense. I must say, however, with regard to the text at hand, I still do not see how it can be said that the δέησις of a righteous man is is made effectual by an external agent. The fact that the man is righteous may add more weight to his petition, but this still looks like a middle voice to me..